Cans of regular Coke are labeled as containing 12.
Statistics students weighted the content of 10 randomly chosen cans, and found the mean weight to be 12.1.
Assume that cans of Coke are filled so that the actual amounts are normally distributed with a mean of 12 and a standard deviation of 0.09. Find the probability that a sample of 10 cans will have a mean amount of 12.1 at least .
I would've used the CLT approximation, but that's only if the sample was greater than 30 cans, so now I don't know what to do! HELP PLEASE!!
To solve this problem, we can use the Central Limit Theorem (CLT) approximation even though the sample size is less than 30. The CLT states that for a random sample of n observations drawn from any population, the distribution of the sample mean will approach a normal distribution as n increases, regardless of the shape of the population distribution.
First, let's define the variables:
μ = population mean = 12 (mean weight of a can of Coke)
σ = population standard deviation = 0.09 (standard deviation of the weights)
n = sample size = 10 (number of cans in the sample)
x̄ = sample mean weight = 12.1
Now, we need to calculate the standard error of the sample mean. The standard error (SE) is equal to the population standard deviation divided by the square root of the sample size:
SE = σ / √n
SE = 0.09 / √10
SE ≈ 0.02846
Next, we need to find the z-score, which measures how many standard errors the sample mean is from the population mean. The z-score is calculated using the formula:
z = (x̄ - μ) / SE
z = (12.1 - 12) / 0.02846
z ≈ 3.5155
Now, we want to find the probability that the sample mean is at least 12.1. This is equivalent to finding the area under the normal curve to the right of the z-score.
Using a standard normal distribution table or a calculator, we can find the probability associated with the z-score of 3.5155. In this case, the probability is extremely small, close to 0.
Therefore, the probability that a sample of 10 cans will have a mean amount of 12.1 or more is very low. This suggests that it is unlikely to obtain a sample mean as high as 12.1 if the actual population mean is 12, assuming the cans of Coke are filled normally.